Question

How to determine if a vector lies in a span of other vectors?

Answer 1

$\mathbf c$ lies in $\mathrm{span}\{\mathbf a,\mathbf b\}$ if $\mathbf c$ can be written as a linear combination of $\mathbf a$ and $\mathbf b$.

In other words, $\mathbf c\in\mathrm{span}\{\mathbf a,\mathbf b\}$ if there exists scalars $k_1,k_2\in\mathbb R$ such that

$\mathbf c=k_1\mathbf a+k_2\mathbf b$

Vectors in $\mathbb R^n$ are equal if their components are equal:

$\langle4,1,0\rangle=k_1\langle1,3,4\rangle+k_2\langle4,9,9\rangle$

$\langle4,1,0\rangle=\langle k_1+4k_2,3k_1+9k_2,4k_1+9k_2\rangle$

$\implies\begin{cases}k_1+4k_2=4\\3k_1+9k_2=1\\4k_1+9k_2=0\end{cases}$

This system has no solutions, so $\mathbf c$ does not belong to $\mathrm{span}\{\mathbf a,\mathbf b\}$.